COMPLEMENTARITY ANALYSIS IN MULTINOMIAL

Transcription

1 1 / 25 COMPLEMENTARITY ANALYSIS IN MULTINOMIAL MODELS: THE GENTZKOW COMMAND Yunrong Li & Ricardo Mora SWUFE & UC3M Madrid, Oct 2017

2 2 / 25 Outline 1 Getzkow (2007) 2 Case Study: social vs. internet interactions 3 The gentzkow command in Stata 4 Results

3 3 / 25 Getzkow (2007) Getzkow (2004): Valuing new goods in a model with complementarity: Online newspapers, The American Economic Review, 2007

4 4 / 25 Getzkow (2007) Online vs. printed news Do online newspapers crowd out consumption of printed newspapers? Print and online papers are substitutes. The welfare benefits of the online newspaper outweigh its costs. The newspaper firm (the Washington Post) gains with advertising. The potential firm s gain of charging a positive price for online content disappears with advertising Methodologically: Many important economic questions hinge on the extent to which new goods either crowd out or complement consumption of existing products. Standard methods for studying new goods rule out complementarity by assumption, so their applicability to these questions has been limited. Getzkow develops a new model that relaxes this restriction.

5 Getzkow (2007) Gentzkow s Multinomial Model u 0 = 0 u A = βa 0 + β A x α A p A + v A u B = βb 0 + β B x α B p B + v B u AB = u A + u B + Γ Definition Alternatives ( ) A and B are complements (substitutes) if and only if sign Q A p B < (>)0 Theorem Alternatives A and B are complements (substitutes) if and only if Γ > (<)0 5 / 25

6 6 / 25 Getzkow (2007) Remarks Getzkow s pattern of complementarity depends on parameter Γ: it describes by how much utility increases when both alternatives are chosen together If the price of one alternative decreases, then the desirability of the other alternative indirectly increases through Γ The complementarity pattern is unrelated to the IIA assumption The model imposes a rank condition on the variance-covariance matrix in a typical multinomial model that can be tested using a LM test: v AB = v A + v B If this condition is not satisfied, then the sign of Γ does not necessarily reflect the complementarity pattern

7 Getzkow (2007) Gentzkow s analysis in the unrestricted multinomial model u 0 = 0 u A = βa 0 + β A x α A p A + v A u B = βb 0 + β B x α B p B + v B u AB = βab 0 + β AB x + α A p A α B p B + v AB Complementarity is individual specific: it depends on x and on (v AB v A v B ). Take δ (x) = ( βab 0 β0 A B) β0 + (βab β A β B ) x + (v AB v A v B ). No link between δ (x) and E We need to numerically estimate E ( ) We can condition on x: E Q A p B x ( ) Q A p B ( ) Q A p B 7 / 25

8 8 / 25 Case Study: social vs. internet interactions Case Study: social vs. internet interactions

9 9 / 25 The data Case Study: social vs. internet interactions We use data from the 2010 national baseline survey of the China Family Panel Studies (CFPS) The sample is representative of 95% of the Chinese population in 2010 Sample: children who use internet between 10 and 15 years old (885 observations) Variables of interest: socialout: the child does at least one of the following activities with friends at least once a month: going to karaoke, going to disco, dining out, and playing at a internet bar. socialnet: one if child says doing social interactions on internet is important for him/her. Prices: for real social activities: pocket money per month, for internet social activities: whether parents use a popular online social interaction application (QQ)

10 Our goals Case Study: social vs. internet interactions In our application: To study whether internet social interactions crowd out real social interactions We estimate the general multinomial model and compute cross-price elasticities In particular, we want to see the role of parents income and child age We need to develop a command for Stata 10 / 25

11 11 / 25 The gentzkow command in Stata The gentzkow command in Stata

12 12 / 25 The gentzkow command in Stata What the ado file does First, it estimates a multinomial model Then, it estimates the partial derivative of the demand of one of the alternatives when a price of the other alternative has an infinitesimal change it is an integral over the utility shocks space: we implement it using a numerical Monte Carlo approximation this object is independent of the alternative AND the price chosen but interpretation is not clear when no price is included in the specification Partial derivatives can be computed for different values of controls, x Standard errors are computed using the bootstrap In the current implementation, mlogit, mprobit, and gsem are available to estimate the multinomial model

13 13 / 25 The gentzkow command in Stata Inputs Required: list of variables: at the very least the two binary choices, possibly expanded with controls x and prices p Optional: number of replications in numerical integration number of bootstrapped samples seed to control random number generation expected obs. per cell under uniform assumption: this is the parameter we use to control the accuracy of the Monte Carlo computation of the integral number of quadrature points (only with mprobit and gsem) control variable whose value changes, and different evaluation values model: mlogit, mprobit, gsem, and all options related to these commands, including constraints whether multinomial model estimates and elasticities are displayed

14 14 / 25 The gentzkow command in Stata Syntax syntax varlist(min=2) [if] [in] [fw pw iw], /// [ /// Replications(integer 500) /// // # replications in integration Bootstraps(integer 100) /// // # bootstrapped samples SEED(real 1966) /// // seed of random number generation Ncells(integer 15) /// // expected obs. per cell INTPoints(integer 15) /// // # of quadrature points Conditioning(varname) /// // evaluation variable VALues(numlist) /// // values for evaluation ATVALues(name) /// // matrix name of control values for evaluation Model(string) /// // mlogit, mprobit, gsem (default is mlogit) NOCONStant /// // suppress constant term CONSTraints(numlist) /// // list of linear constraints Level(real 95) /// // set confidence level; default is level(95) Detail /// // displays multinomial model estimates CROSS /// // only shows cross elasticities MAXimize(string) /// // string containing maximize_options TWOlevel(varname) /// // variable for level random effects (gsem) * ]

15 15 / 25 Example 1 (I) The gentzkow command in Stata gentzkow socialout socialnet prices lista_x0 Dp*, /// constraint(1/ i ) /// r(5000) b(200) n(50) /// max("iterate(50)") /// cross constrained multinomial logit bootstrap standard errors using 200 simulations numerical integration using 5000 simulations, with around 50 observations within each cell to compute frequencies only displays cross-price elasticity

16 Example 1 (II) The gentzkow command in Stata (running mlogit on estimation sample) Bootstrap replications (200) Warning: only 191 converged bootstraps Cross-price elasticity Coef. Std. Err. z P> z [95% Conf. Interval] _cons / 25

17 17 / 25 Example 2 (I) The gentzkow command in Stata gentzkow socialout socialnet prices lista_x0 Dp*, /// constraint(1/ i ) /// r(5000) b(200) n(50) /// max("iterate(50)") /// c(age) val(10(2)14) /// cross test _b[r1]-_b[r3]=0 computes expectation conditional on age: 10,12,14 only displays cross-elasticities tests whether the expectation is equal at age 10 than at age 14

18 Example 2 (II) The gentzkow command in Stata (running mlogit on estimation sample) Bootstrap replications (200) Cross-price elasticity Coef. Std. Err. z P> z [95% Conf. Interval] r r r ( 1) r1 - r3 = 0 chi2( 1) = 7.78 Prob > chi2 = / 25

19 19 / 25 Results Results

20 20 / 25 Results Cross elast. Std.Err. Own price: Real life Std.Err. Own price: Internet Std.Err. Model,Elasticities Unconditional (0.008) *** (0.009) *** (0.010) Uncond., Province ** (0.014) *** (0.022) *** (0.018) Conditional (0.013) *** (0.015) *** (0.017) Cond., Province *** (0.012) *** (0.022) *** (0.017) Cond., Major Prov *** (0.011) *** (0.021) *** (0.015) Note: Estimated cross-price elasticities and own-price elasticities of our unrestricted multinomial model using the multinomial logit model option are reported in the table. Cross elast. refers to the change in the child s probability of participating in real life social interactions with respect to an infinitesimal change in a factor that positively affects the child s net utility of doing internet social interactions but does not affect the child s net utility of doing real life social interactions. Equivalently, Cross elast. also refers to the change in the child s probability of participating in internet social interactions with respect to an infinitesimal change in a factor that positively affects the child s net utility of doing real life social interactions but does not affect the child s net utility of doing internet social interactions. Std.Err. refers to the standard error. Own price: Real life refers to the change in the child s probability of participating in real life social interactions with respect to an infinitesimal change in a factor that positively affects the child s net utility of doing real life social interactions but does not affect the child s net utility of doing internet social interactions. Own price: Internet refers to the change in the child s probability of participating in internet social interactions with respect to an infinitesimal change in a factor that positively affects the child s net utility of doing internet social interactions but does not affect the child s net utility of doing real life social interactions. Note: Prices affect utility positively.

21 21 / 25 Results P-value of test Value Cross elast. Std.Err. Value Cross elast. Std.Err. A. Price variables Pocket money of the child in *** (0.007) *** (0.018) log Both parents use QQ everyday *** (0.011) * (0.011) B. Child characteristics Gender ** (0.010) *** (0.014) Age in years (0.006) *** (0.016) C. Household characteristics Urban ** (0.010) *** (0.014) Note: Estimated cross-price elasticities of our unrestricted multinomial model using the multinomial logit model option are reported in the table. The multinomial logit model is estimated with price variables, all the other controls, and major province dummies. Cross-price elasticities are estimated for a series of values of each of the controls. P-value of test refers to the p-value of the test on the joint equality of all the cross-price elasticities based on different values of the controls. Value refers to different values of the control variable. Cross elast. refers to the change in the child s probability of participating in real life social interactions with respect to an infinitesimal change in a factor that positively affects the child s net utility of doing online social interactions but does not affect the child s net utility of doing real life social interactions. Equivalently, Partial derivative also refers to the change in the child s probability of participating in online social interactions with respect to an infinitesimal change in a factor that positively affects the child s net utility of doing real life social interactions but does not affect the child s net utility of doing online social interactions. Std.Err. refers to the standard error.

22 22 / 25 Results Internet and traditional social interactions Complementarity by income level Monthly income of the parents (1000 yuan)

23 23 / 25 Results Internet and traditional social interactions Complementarity by child age Age in years of the child

24 24 / 25 Extensions Extensions coefficient s display should be improved only cross-elasticity results posted as results, all results stored in matrix panel data implementation not checked out: Monte Carlo using gsem testing the Gentzkow model using LM test

25 25 / 25 Extensions Thank you